Lower Partial Moments under Gram Charlier Distribution:

Performance Measures and Efficient Frontiers∗

Angel Leon†

Univ. of Alicante

Manuel Moreno‡

Univ. of Castilla-La Mancha

February 22, 2015

Abstract

We derive closed-form expressions for the performance measures (PMs) based onthe lower partial moments (LPMs), such as the Farinelli-Tibiletti and Kappa measures,with Gram-Charlier (GC) density for returns. It is verified that the LPMs can beobtained as a linear function on both higher moments, skewness and excess kurtosis.We also show that these PMs influence differently to the Sharpe ratio in rankingportfolios due to the effects of the higher moments. We also obtain the efficientfrontiers (EFs) based on the mean-LPM framework. We find important differencesbetween portfolios from different EFs regarding their stock compositions, portfolioskewness and excess kurtosis levels. Finally, we also obtain closed-form expressionsfor PMs under a more flexible density like the SNP density which nests the GC density.

JEL Classification: C10, C61, G11, G17.Keywords: Downside risk, Performance measure, Rank correlation, Efficient frontier,Co-lower partial moment matrix, Copula, SNP distribution.

∗The authors are very indebted with Walter Paiva for a primer version of this paper. We would alsolike to acknowledge the very helpful comments from Lluis Navarro, Belen Nieto, Gonzalo Rubio, FranciscoSogorb and Antonio Falco. The contents of this paper are the sole responsibility of the authors.

†Angel Leon Valle (corresponding author) is from Department of Quantitative Methods and EconomicTheory, University of Alicante, San Vicente del Raspeig, 03080, Alicante, Spain. Phone: +34 96 5903400,ext. 3141, fax: +34 96 590 3621. E-mail: aleon@ua.es. Angel Leon would like to acknowledge the financialsupport from the Spanish Ministry for Science and Innovation through the Grant ECO2011-29751, andalso, from the Grant PrometeoII/2013/015.

‡Manuel Moreno is from Department of Economic Analysis and Finance, University of Castilla La-Mancha, Cobertizo San Pedro Martir s/n, 45071 Toledo, Spain. Phone: +34 925268800, ext. 5133, fax:+ 34 925268801. Manuel Moreno acknowledges the financial support from the grants P08-SEJ-03917,ECO2012-34268, and PPII-2014-030-P.

1

1 Introduction

Whether the popular Sharpe ratio (SR), proposed by Sharpe (1966, 1994), is an ade-quate performance measure (reward-to-variability index) for ranking financial productsstill remains a controversial question among academics and practitioners. Although SRis fully compatible with normally distributed returns, and more general with ellipticaldistributions1 of returns, it may lead to incorrect evaluations when stock returns exhibitasymmetry and / or heavy tails. Since the standard deviation as the two-sided variabilitymeasure in SR seems to be dubious as a risk measure, several one-sided type measures ofrisk have been proposed. Their corresponding performance measures (PMs) are known asone-sided PMs depending on a portfolio return threshold, τ . Some of these PMs are alsocharacterized by one-sided type reward measures.

The one-sided PMs are mainly the following two groups: the Generalized Rachev ratio(GRR) based on the conditional VaR (CVaR), see Biglova et al. (2004), and the Farinelli-Tibiletti (FT) ratio based on the partial moments (upper and lower partial moments,respectively, for the reward and risk measures), see Farinelli and Tibiletti (2008). Bothgroups use one-sided measures to modeling the reward and risk measures. Finally, anotherperformance measure known as the Sortino and Satchell (SS) ratio (2001), or Kappa ratio,verifies that only the risk measure is a one-sided measure (lower partial moment). Ourwork aims, among others, at the behaviour of those PMs based on partial moments undera static portfolio performance and assuming a certain portfolio return distribution soas to get closed-form expressions. More alternative reward-to-variability ratios are welldocumented in Caporin et al. (2013) and their references inside.

There is a stream of research concluding that the choice of the performance measuredoes not affect the ranking in portfolios. Eling and Schuhmacher (2007) and Eling (2008)find that rank correlations between SR and alternative PMs for hedge fund (HF) data arehighly positively correlated. These empirical results are also supported by some theoreticalstudies such as Schuhmacher and Eling (2011, 2012). They show that under severalconditions, the PM is a strictly increasing function in SR. Nevertheless, we show thatrank correlations from FT measures and SR exhibit low rank correlations in most cases.Our results agree, among others, with the empirical evidence in Eling et al. (2011).

There are also some recent studies, based on a dynamic portfolio performance setting,concluding that the one-sided PMs outperform the benchmark SR. For instance, Biglovaet al. (2004) and Farinelli et al. (2008, 2009) show that the cumulative final wealth ishigher by implementing either GRR or FTR instead of SR. Another example is Caporinand Lisi (2011), who analyze the dynamic properties of rank correlations between PMsand SR based on a rolling window analysis. They find low rank correlations for some PMsfrom the FT group and so on. In short, this empirical evidence suggests that the selectionof the performance measure matters.

We also study the rank correlation sensitivity when returns departure from the nor-mality assumption (skewness, s, and / or excess kurtosis, ek, different from zero). Wecan find some PMs from the SS group (or Kappa measures), which are highly correlated

1See, for more datails, Owen and Rabinovitch (1983).

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with SR when considering the whole sample. Nevertheless, if we divide the total sampleof funds between a group with lower SR and another with higher SR values, it is verifiedthat rank correlations tend to decrease for those funds from the higher SR group. Theseresults can be explained through the levels of s and ek implied in the distributions of theseportfolio returns. This evidence is also confirmed by Zakamouline (2011).

This paper has several goals. First, we try to understand the behaviour of PMs whenthe portfolio returns departure from the normal distribution and so, how important theselection of alternative PMs (against SR) can be in ranking portfolios. We obtain closed-form expressions of PMs based on the lower partial moments (LPMs), proposed by Bawa(1975) and Fishburn (1977),2 as downside risk measures. We mainly concentrate on thoseone-sided PMs from the FT3 and SS groups. We assume that the probability densityfunction (pdf) for the portfolio returns is driven by the Gram-Charlier (GC) expansion.We also get the closed-form expression for the GC shortfall probability to obtain somePMs with downside risk measures based on the value at risk (VaR), such as both theReward-to-VaR and the Reward-to-CVaR.

The GC distribution has been implemented, among others, by Corrado and Su (1996),Jondeau and Rockinger (2001) and Jurczenko and Maillet (2006). The advantage of theGC distribution over alternative distributions is that the higher moments s and ek directlyappear as the pdf’s parameters. Jondeau and Rockinger (2001) obtain the constraintsthat s and ek must hold in order to guarantee a well-defined pdf. We get easy closed-form expressions for the LPMs, which can be expressed as simple linear functions on boths and ek. Hence, we can easily understand the behaviour of these risk measures dueto changes of s and ek. For instance, Passow (2005) obtains closed-form expressions, inparticular, for the Sharpe-Omega ratio (or Kappa measure of order one) under the Johnsondistribution (JD) family. Contrary to the GC pdf, the unbounded Johnson distributionfrom the JD family allows for higher levels for both s and ek. This result becomes animportant advantage with respect to the GC distribution but it has the drawback thatthe Sharpe-Omega ratio under the JD family becomes more cumbersome and difficult tointerpret for changes of s and ek than in our case.

The restriction of the GC pdf to capture some levels of s and ek of portfolio returnssuggests other distributions trying to seize better these higher moments but getting morecomplex or hard tractable expressions for either the FT or SS measures under differentdistributions. Nevertheless, as a possible solution to the restricted higher moments underGC, we suggest and obtain the LPM closed-form expresions under the SNP density, intro-duced by Gallant and Nychka (1987), whose parametric properties have been studied byLeon et al. (2009).4

Second, we obtain the efficient frontiers (EFs) based on the LPMs. We compare thecomposition of the portfolios from the EFs under both the mean-LPM and the Markowitz

2Some other seminal references about LPMs are Bawa and Lindenberg (1977), Holthausen (1981),Harlow and Rao (1989) and Harlow (1991).

3Note that the reward measures from the FT ratios are just the upper partial moments, which can beexpressed in terms of the LPMs.

4The SNP pdf allows a higher flexibility than GC in terms of s and ek. Its density is always positiveand it nests the GC distribution.

3

(1959) mean-variance (MV) settings. For the minimization of the LPM of a portfolioreturn, we use the symmetric co-lower partial moment matrix from Nawrocki (1991). Forthe comparison between portfolio frontiers, we use the root mean squared dispersion index(RMSDI), see Grootveld et al. (1999). A sensitivity analysis is implemented by changingthe return threshold, τ .

We also obtain the theoretical skewness and excess kurtosis levels implied in the port-folios from these EFs. Note that it is necessary to get previously both (theoretical) co-skewness and co-kurtosis matrices by assuming in this paper a Gaussian copula for thedependence among the different stock returns and the GC distribution for their marginaldistributions. Our multivariate setting for stock returns agrees with the procedure tocompute EFs under the mean-LPM (MLPM) approach by Nawrocki (1991). We find dif-ferences when obtaining both s and ek from a frontier portfolio under the MV approachagainst another frontier portfolio based on the LPM measure.

The rest of the paper is organized as follows. In Section 2 we present some portfolioperformance measures based on the LPMs, that is, both FT and SS (or Kappa) measures.In Section 3 we show the GC pdf, implemented by JR (2001), for the behaviour of returnsand some characteristics of this density. In Section 4 we obtain the closed-form expressionfor the LPM measure of order 0, which is also known as the shortfall probability. Wealso analyze here the behaviour of the shortfall probability under different values of τ . InSection 5 we can easily obtain the closed-form expressions for LPM measures under theGC distribution and hence, the corresponding FTR and Kappa measures. We show thebehaviour of the Kappa ratios regarding the level of s and ek. We also get the iso-curves,mainly for the Kappa measures.

In Section 6 we conduct an intensive simulation study using the GC distribution forthe performance evaluation. In Section 7 we get the EFs by using the LPMs under theGC density and compare with the traditional MV frontier. We also obtain both s andek from the MLPM efficient frontiers in order to study the behaviour of them. Section 8shows the SNP distribution and the closed-form expressions of LPMs. Finally, Section 9summarizes and provides the main conclusions. The description of our data series for oursimulation studies can be found in Appendix A. The proofs of propositions and corollariesare deferred to Appendix B.

2 Performance measures based on partial moments

The lower partial moments (LPMs) measure risk by negative deviations of the stock re-turns, r, in relation to a minimal acceptable return, or return threshold, τ . Fishburn(1977), among others, defines the LPM of order m for a stock return as

LPMf (τ,m) = Ef [(τ − r)m+ ] =

∫ τ

−∞(τ − r)mf(r) dr, (1)

where f (·) denotes the probability density function (pdf) of r and (y)+=max (y, 0). AsLPMs consider only negative deviations of returns from τ , they seem to be a more ap-propriate measure of risk than the standard deviation, which considers both negative andpositive deviations from the expected return µ. Hence, returns below the threshold, τ ,

4

are seen by investors as loss. This (loss) threshold level, τ , might be the rate of inflation,the real interest rate, the return on a benchmark index, a risk-free rate, etc. About theparameter m, LPM with the order 0 < m < 1 can express ’risk seeking’, for m = 1 ’riskneutrality’, and m > 1 ’risk aversion’ behaviour for the investor. Thus, the higher m themore risk averse an investor. The LPMs of order 0 and 1 are, respectively, the shortfallprobability and the expected shortfall. Finally, the upper partial moment (UPM) is definedas

UPMf (τ, q) = Ef [(r − τ)q+] =

∫ −∞

τ(r − τ)qf(r) dr. (2)

Note that we can express UPMf (τ, q) as a function of LPMf (τ, q) and hence, we can speakabout performance measures based on LPMs. Moreover, some performance measuresbased on the basis of the Value at risk (VaR) can also be expressed in terms of the LPMs,as shown at the end of this section.

Throughout this paper, we will obtain closed-form PM expressions by assuming theGram-Charlier (GC) pdf for the standardised portfolio returns, z, and setting m and qfrom (1) and (2) to be integer numbers, i.e. m, q ∈ N+ ≡ {1, 2, 3, ...}.

2.1 The Kappa measures

The corresponding risk-adjusted return PMs based on LPMs are the SS or Kappa ratios,see Sortino and Satchell (2001), which are defined as

Kf (τ,m) =µ− τ

m√LPMf (τ,m)

, (3)

where µ is the mean of the (portfolio) stock return r with f (r) as pdf, i.e. µ=Ef [r]. Notethat µ− τ is just an excess expected return.5 Some popular measures which are nested in(3) are the Omega-Sharpe ratio (see Kaplan and Knowles, 2004) for m = 1, the Sortinoratio (see Sortino and van der Meer, 1991) for m = 2 and Kappa 3 (see Kaplan andKnowles, 2004) for m = 3.

2.2 The Farinelli-Tibiletti ratio (FTR)

Next, we show an alternative performance measure (see Farinelli and Tibiletti, 2008)known as the Farinelli-Tibiletti ratio (FTR), which nests a family of PMs dependingon both the LPMs in (1) and the UPMs in (2). The FTR is defined as

FTRf (τ, q,m) =q√UPMf (τ, q)

m√LPMf (τ,m)

, (4)

where q, m > 0. It is known that the higher q and m, the higher the agent’s preferencefor (in the case of expected gains, parameter q) or dislike of (in the case of expected losses,parameter m) extreme events. Note that (4) nests two popular PMs: (i) the Omega ratio6

(see Keating and Shadwick, 2002) when q = m = 1, (ii) the Upside Potential ratio (see

5This is a more general definition of excess expected return. If τ = rf , where rf denotes the risk-freerate, we have the more usual expression of excess expected return.

6It is shown later that Kf (τ, 1) = FTRf (τ, 1, 1) − 1. Finally, the Bernardo and Ledoit (2000) ratio isjust the Omega ratio for τ = 0 and hence, it represents the gain-to-loss ratio.

5

Sortino et al., 1999) when q = 1 and m = 2. Rewritting (2) as a function of (1), we havean alternative expression of FTR in the following corollary.

Corollary 1 Let ψf (τ, q)=Ef [(r− τ)q] and f (·) be the pdf of the portfolio stock return r,then (4) can be expressed as

FTRf (τ, q,m) =

q

√ψf (τ, q) + (−1)q+1 LPMf (τ, q)

m√LPMf (τ,m)

. (5)

Proof. It is straightforward.

Finally, we can also rewrite (5) in terms of the Kappa measures (3). Thus,

FTRf (τ, q,m) =

q

√ψf (τ, q) + (−1)q+1 (µ− τ)qKf (τ, q)−q

(µ− τ)Kf (τ,m)−1. (6)

Note that for q = 1 in (6), we have ψf (τ, 1) = µ − τ and then, FTRf (τ, 1,m) =Kf (τ,m)

[1 +Kf (τ, 1)

−1].

2.3 Performance measures on the basis of VaR

The Value at risk is just the α-quantile of the distribution of the portfolio return r withf (·) as pdf and denoted as V aRα

f . Since LPMf (V aRαf , 0) = α, we can get V aRα

f by theinversion method. Connecting with this risk measure, we have as performance measurethe Reward-to-VaR (see Dowd, 2000):

RV aRαf (τ) =

µ− τ∣∣∣V aRαf

∣∣∣, (7)

where |·| denotes the absolute value function.

Another risk measure is the Expected Shortfall (ES), i.e. ESαf = − 1

αEf

[r∣∣∣r ≤ V aRα

f

].

It is also known as the conditional value at risk, CV aRαf . We can relate LPMf (τ, 1)

to ESαf with τ = V aRα

f and so, we obtain7 LPMf (V aRαf , 1) = αESα

f + αV aRαf . The

connected performance measure with CV aRαf (or ESα

f ) is known as the Reward-to-CVaR(see Agarwal and Naik, 2004):

RCV aRαf (τ) =

µ− τ

CV aRαf

=µ− τ

(1/α)LPMf (V aRαf , 1)− V aRα

f

. (8)

3 GC density and properties

The stock return r is a random variable defined as

r = µ+ σz, z ∼ GC(0, 1, s, ek), (9)

7See Jarrow and Zhao (2006).

6

such that the pdf of the random variable z is the Gram-Charlier expansion with zero mean,unit variance, s and ek as the levels, respectively, of skewness and excess kurtosis. Hence,the return in (9) is just an affine transformation of a random variable with GC expansionas pdf. The GC pdf denoted as g (z), where z is a standardised random variable,8 isdefined as

g (z) = p (z)φ (z) ; p(z) = 1 +s√3!H3(z) +

ek√4!H4(z), (10)

where φ (·) denotes the pdf of the standard normal variable and Hk (z) is the normalizedHermite polynomial of order k. These polynomials can be defined recursively for k ≥ 2 as

Hk(z) =zHk−1(z)−

√k − 1Hk−2(z)√k

, (11)

with initial conditions H0 (z) = 1 and H1 (z) = z. It holds that {Hk (z)}k∈N constitutesan orthonormal basis with respect to the weighting function φ(z), that is,

Eφ[Hk(z)Hl(z)] = 1(k=l), (12)

where 1(·) is the usual indicator function and the operator Eφ[·] takes the expectation ofits argument regarding φ (·). Note that Eφ [Hk (z)] = 0 for k ≥ 1 and then, (12) is justthe covariance between Hk (z) and Hl (z).

3.1 Positivity of pdf

Note that g (z) in (10) allows for additional flexibility over a normal density since it in-troduces both skewness and (excess) kurtosis of the distribution. It can lead to negativevalues for certain values of both centered moments. JR (2001) obtain numerically a re-stricted space Γ for possible values of (ek, s) to guarantee the positivity of g (z). Theconstrained GC expansion restricted to Γ will be referred as the true GC density. Figure1 exhibits Γ with a frontier (the envelope) delimiting the oval domain. Γ is a compact andconvex set. Note that ek ∈ [0, 4] while |s| ≤ 1.0493 verifying that the range of s dependson the level of ek. For instance, if |s| = 0.6 then ek ranges from 0.6908 to 3.7590, while fors = 0, ek ranges from 0 to 4. The maximum size for skewness is reached for ek = 2.4508.Obviously, the case for the normal distribution corresponds to the origin.

[ INSERT FIGURE 1 AROUND HERE ]

Gallant and Tauchen (1989) suggest to square the polynomial p (·) in (10) to guaranteea true pdf under the CG framework. In this case, we lose the interpretation of somemoments of the distribution. Leon et al. (2009) analyze the statistical properties of semi-nonparametric (SNP) distributions, which were introduced by Gallant and Nychka (1987).The SNP density (always positive by construction) can be expressed as a Gram-Charlierseries of Type A, which is the product of a standard normal density times an infinite seriesof Hermite polynomials. So, the GC density in (10) is just a truncated GC expansion Itis shown that Γ is contained into a higher space under SNP distributions.9 Contrary tothe GC density, the parameters implied in the SNP pdf do not correspond directly to the

8The pdf of r in (9) is obtained as f (r) = g (z) /σ.9See the skewness-kurtosis frontiers under the SNP distribution in Figure 1 from Leon et al. (2009).

7

levels of skewness and kurtosis of the distribution. In Section 8 we get the closed-formexpressions for the LPMs under the SNP distribution.

Finally, we select some data consisting in monthly return series for ten hedge fund (HF)index strategies. They are denoted as HFj , j = 1, . . . , 10. See Appendix A for more detailsabout these data. Table 3 in Appendix A exhibits the constrained maximum likelihood(CML) estimation of the parameters implied in (9) and (10)for each monthly series. Fromnow on, these estimations will be used in this paper to implement simulations.

3.2 Moments of r and z

To show that z ∼ GC(0, 1, s, ek) with pdf g (·) in (10), we need to obtain the first four(noncentral) moments of z. These moments can be obtained by using the relationshipbetween the powers of z and the Hermite polynomials in (11) and the condition in (12):

Eg [z ] = Eg [H1 (z)] = 0,

Eg

[z2]

=√2Eg [H2 (z)] + 1 = 1,

Eg

[z3]

=√3!Eg [H3 (z)] + 3Eg [H1 (z)] = s,

Eg

[z4]

=√4!Eg [H4 (z)] + 6

√2Eg [H2 (z)] + 3 = ek + 3.

The following proposition aims to obtain a general expression for Eg

[zk]where k ∈ N+.

Proposition 1 The general expression for Eg

[zk], where k ≥ 5 and pdf g (·) in (10), is

given as

Eg

[zk]=

{λk,0 + λk,1 ek, k is even,

λk,2 s, k is odd,(13)

where λk,l ∈ R can be seen in Appendix B. Since r in (9) is an affine transformation of z,the non-central moments of r are obtained as

Ef [rk] =

k∑

n=0

(k

n

)µk−nσnEg[z

n], (14)

where(kn

)= k!

n!(k−n)! .

Proof. See Appendix B.

Note that, if k is even (odd) Eg

[zk]depends only on the excess kurtosis (skewness).

4 Shortfall probability: LPM(τ, 0)

The LPM equation in (1) for m = 0 is just the distribution function for the standardisedreturn z in (9). The expression LPMf (τ, 0) is also known as the ”shortfall probability”.

Proposition 2 The shortfall probability is given by

LPMf (τ, 0) = Φ (τ∗)− s

3√2!H2 (τ

∗)φ (τ∗)− ek

4√3!H3 (τ

∗)φ (τ∗) , (15)

where τ∗ = (τ − µ) /σ.

8

Proof. See Appendix B.

Remember that getting some performance measures based on VaR, e.g. (7) and (8),we need to compute V aRα

f . As LPMf (V aRαf , 0) = α, we get V aRα

f by the inversion

method. According to Proposition 2, V aRαf = µ + σF−1

GC (α; s, ek).10 Note that, for anormal distribution, we get V aRα

f = µ + σzα, where zα the α-quantile from a N(0, 1)distribution.

The following Corollary shows the behaviour of LPMf (τ, 0) with respect to the pa-rameters s, ek, τ , µ and σ.

Corollary 2 Let LPMf (τ, 0) in (15) and τ∗ = (τ − µ) /σ, it holds that

i) ∂LPMf (τ, 0)/∂s > 0 ⇔ |τ∗| < 1.

ii) ∂LPMf (τ, 0)/∂ek > 0 ⇔ τ∗ ∈(−∞,−

√3)∪(0,√3).

iii) For τ∗ ∈(−√

3−√6, 0)∪(√

3 +√6,+∞

), then ∂LPMf (τ, 0)/∂τ

∗ > 0 and so,

iii.1) ∂LPMf (τ, 0)/∂τ > 0.

iii.2) ∂LPMf (τ, 0)/∂µ < 0.

iii.3) ∂LPMf (τ, 0)/∂σ > 0 iff µ > τ .

Proof. See Appendix B.

As a result, for τ∗ ∈ (0, 1), we have ∂LPMf (τ, 0)/∂s > 0 and ∂LPMf (τ, 0)/∂ek > 0.Next, we study the behaviour of the shortfall probability for different values of τ and(ek, s) ∈ Γ.

4.1 Value at Risk as threshold level

If LPMf (τ, 0) = α, the threshold level τ is just the α-quantile of the distribution of thestock return. Thus, τ = V aRα

f for very small values of α. To start with, take as anexample the HF8 monthly returns whose parameters from Table 3 in Appendix A are:µ = 0.86%, σ = 2.61%, s = 0.4, and ek = 1.52. For these returns, the threshold levelsτ0.01 = −5.49% and τ0.05 = −2.97% correspond, respectively, to the 1% and 5% quantiles.Under the GC distribution, the same α-quantiles for the standardised stock return z areτ∗0.01 = −2.433 and τ∗0.05 = −1.467.

Figure 2 shows the shortfall probability in (15) for different values of (ek, s) ∈ Γsubject to µ = 0.86%, σ = 2.61%. The different lines in this Figure illustrate the possiblecombinations obtained from τ = τ0.01, τ0.05 and s = −0.7, 0, 0.4. Fixing the skewness level,we can see that the lines for τ0.01 (τ0.05) exhibit a positive (negative) slope. This is dueto the result (ii) in Corollary 2. Note also that the length of each line is different becausethe range of ek changes with the level of s for the GC pdf to be well defined.

10Let zGC,α ≡ F−1

GC (α; s, ek) where FGC (·) is another way to denote the LPM for m = 0 in (15). Then,zGC,α is obtained through a numerical search based on the bisection method by using the Matlab function’fzero’. For more details, see Brandimarte (2006).

9

We can see that, for any τ , the upper (lower) line corresponds to s = −0.7 (s = 0.4).This is a consequence of (i) from Corollary 2. That is, the shortfall probability increases(decreases) independently of ek since the skewness level decreases (increases) with respectto, for instance, the initial level of s = 0.11 Two points for HF8 can be seen in twodifferent lines: (1.52, 0.01) in line where s = 0.4, τ = −5.49% and (1.52, 0.05) in the linewith s = 0.4, τ = −2.97%. Any other point can be considered as a hypothetical portfoliowith the same µ and σ as HF8.

Finally, a point from a certain line, independently of s, shows that the higher its eklevel the higher its value of LPMf (τ0.01, 0) while the opposite happens for τ0.05. Moreprecisely, increasing ek and considering s = −0.7 (0.4), LPMf (τ0.01, 0) can increase from2.49% (0.3%) to 4.17% (2.46%) while LPMf (τ0.05, 0) can decrease from 8.31% (5.81%) to6.38% (3.32%) for s = −0.7 (s = 0.4). Now, the reason is (ii) from Corollary 2.

[ INSERT FIGURE 2 AROUND HERE ]

4.2 Non-negative threshold levels

Figure 3 shows the same analysis as Figure 2 for HF8 but now we take non-negative thresh-olds such as τ = 0% and τ = 0.39%. The last one is the risk-free monthly interest rateextracted from 10-year US Treasury bonds obtained as the sample mean from November1998 to February 2008 (i.e., 4.72% per annum).12 We highlight three features from thisFigure. First, every line exhibits a negative slope meaning that the higher ek the lowerthe shortfall probability. Second, contrary to Figure 2, for any τ , the lines with positiveskewness are above those for negative skewness.13 Third, there are two points associatedwith HF8 in two different lines: (1.52, 0.37) in line with s = 0.4, τ = 0% and (1.52, 0.44) inthe line with s = 0.4, τ = 0.39%. Again, any other point is just a different (hypothetical)portfolio with the same µ and σ as HF8.

[ INSERT FIGURE 3 AROUND HERE ]

5 LPMs and related PMs with GC distribution

This section starts providing the general closed-form expressions of LPMf (τ,m) and sothe related expressions for FTRf (τ, q,m) and Kf (τ,m) where the stock return is drivenby (9). We show how these LPMs are linear functions of both s and ek, and the behaviourof the above PMs. We also obtain some Kappa (FT) iso-curves.

5.1 Closed-form expressions

Proposition 3 Let z be the standardised return of r in (9). The lower partial moment oforder m ∈ N+ for the security return r can be expressed as

LPMf (τ,m) = LPMn(τ,m) +s√3!θ2,m +

ek√4!θ3,m, (16)

11Note that (i) from Corollary 2 holds, in our case, that ∂LPMf (τ, 0)/∂s < 0 since |τ∗k | > 1 for

k = 0.01, 0.05.12This period is the same as the HF database in Appendix A.13Note that (i) from Corollary 2 holds now that ∂LPMf (τ, 0)/∂s > 0 since the values of τ∗ = (τ −

0.86%)/2.61% for τ = 0%, 0.39% verify that |τ∗| < 1.

10

where LPMn(τ,m) is the LPM by assuming the normal distribution for r in (9), denotedas n (·), with

LPMn(τ,m) =∑m

k=0(−1)k

(m

k

)(τ − µ)m−k σkBk, (17)

and

θj,m =∑m

k=0(−1)k

(m

k

)(τ − µ)m−k σkAkj , (18)

such that Bk = Bk (τ∗) and Akj = Akj (τ

∗), with τ∗ = (τ − µ) /σ, can be seen in AppendixB.

Proof. See Appendix B.

As can be seen, the behaviour of θj,m depends on the expected return, the volatilityand the return threshold, i.e. θj,m = θj,m (µ, σ, τ). Note that θ2,m and θ3,m measure,respectively, the sensitivity of LPMf (τ,m) to changes in s and ek. Note that the Kappameasures are easily obtained by using (3) and (16). The following Corollary shows thegeneral expression for the FT measures in (5) under the GC density for the standardisedreturn.

Corollary 3 Let z be the standardised return of r in (9). The performance measureFTRf (τ, q,m) in (5) for q,m ∈ N+ can be expressed as

FTRf (τ, q,m) =

q

√ψf (τ, q) + (−1)q+1 LPMf (τ, q)

m√LPMf (τ,m)

, (19)

where LPMf (τ, ·) is given in (16) and ψf (τ, q)=Ef [(r − τ)q] is obtained as

ψf (τ, q) =∑q

k=0

(q

k

)(µ− τ)q−k σkEg

[zk], (20)

where Eg

[zk]is given in (13).

Proof. It is straightforward.

As many studies about performance evaluation focus on some popular Kappa measures,such as the Omega-Sharpe, Sortino and Kappa 3 ratios, and the Upside Potential ratiofrom the FT measures, we are very interested in the expression of LPMf (τ,m) for m =1, 2, 3 shown in the following Corollary.

Corollary 4 The expressions of θj,m for j = 2, 3 and LPMn(τ,m) for m = 1, 2, 3 in (16)are:

θj,1 = (τ − µ)A0j − σA1j ,

θj,2 = (τ − µ)2A0j − 2(τ − µ)σA1j + σ2A2j , (21)

θj,3 = (τ − µ)3A0j − 3(τ − µ)2σA1j + 3(τ − µ)σ2A2j − σ3A3j ,

11

and

LPMn(τ, 1) = (τ − µ)Φ (τ∗) + σφ (τ∗) ,

LPMn(τ, 2) = (τ − µ)2 Φ (τ∗) + (τ − µ) σφ (τ∗) + σ2Φ (τ∗) , (22)

LPMn(τ, 3) = (τ − µ)3 Φ (τ∗) + (τ − µ)2 σφ (τ∗) + 3 (τ − µ)σ2Φ (τ∗) + 2σ3φ (τ∗) .

where the values for Akj = Akj (τ∗) with τ∗ = (τ − µ) /σ, can be seen in the Appendix

B.14

Proof. See Appendix B.

5.2 Behaviour of Kappa measures respecting s and ek

We analyze the effects of the higher moments on the performance ratios. We fix theparameter vector (µ, σ, τ) at (µ0, σ0, τ0). Then, LPMf (τ0,m)=LPMm is a function, gm,on both s and ek. Let ∆LPMm and dLPMm denote, respectively, the increment and thetotal differential of LPMm with respect to its arguments.15 The next Corollary inmediatelyarises.

Corollary 5 If we approximate ∆LPMm by dLPMm, we get

∆LPMm =∂gm∂s

∆s+∂gm∂ek

∆ek

holding that

∆LPMm > 0 ⇔ ∆s < ϕm∆ek, ϕm = − θ3,m2θ2,m

. (23)

Proof. It is straightforward.

Table 1 exhibits the behaviour of the popular Kappa measures by changing either sor ek, that is, we provide these measures for alternative portfolios with the same µ andσ but different values for s and ek such that (ek, s) ∈ Γ. We consider again the valuesrelated to HF8 (µ = 0.86%, σ = 2.61%), τ = rf (i.e., 0.39%), and three possible values ofskewness (s = −0.7, 0, 0.4). The Sharpe ratio, SR = (µ− rf ) /σ, is equal to 0.1796 and itis constant across this Table. The SR is considered as the benchmark measure. Pluggingthese parameters into the expression for ϕm in (23), we get ϕ1 = −1.3471, ϕ2 = 0.0499and ϕ3 = 0.2289.

[ INSERT TABLE 1 AROUND HERE ]

The columns 2 to 10 show the three Kappa measures for the different point combina-tions (ek, s) according to different levels of skewness. The main results can be summarizedas follows.

14Note that LPMn(τ,m) in (22) could also be denoted, to shorten, as θ1,m for the case of j = 1 in (21)since both expressions coincide. Nevertheless, we have decided in this paper to use LPMn(τ,m) insteadof θ1,m. Thus, we denote the coefficients of s and ek in (16), respectively, as θ2,m and θ3,m.

15That is, ∆LPMm = gm (s+∆s, ek +∆ek)− gm (s, ek), where ∆x represents a small increment in x.

12

1. Let be the portfolio π1, with (ek, s) = (0.8996, 0), and build new portfolios by onlyincreasing ek. We can see that, if ek increases,Kf (rf , 1) increases but bothKf (rf , 2)and Kf (rf , 3) decrease.

2. Take a new portfolio π2 with (ek, s) = (0.8996, 0.4). It holds that Kf (rf ,m) in-creases if we only increase s by changing π1 for π2. The same behaviour holds foralternative values of ek.

3. Suppose now that ek increases and s decreases. Consider either portfolio π3, with(ek, s) = (1.2048,−0.7), or π4, with (ek, s) = (2.1205,−0.7). It is verified that bothKf (rf , 2) and Kf (rf , 3) decrease when going from π1 to either π3 or π4. Meanwhile,there are opposite effects about the behaviour of LPMf (rf , 1). Note that Kf (rf , 1)decreases for π3 while it increases for π4.

4. Finally, we can see that (23) holds under these examples and so, the behaviour oftheir related Kappa measures is verified. Thus, changing π1 for π2 leads to ∆s = 0.4,∆ek = 0. The case of changing π1 for π3 leads to ∆s = −0.7, ∆ek = 0.3052. Finally,changing π1 for π4 leads ∆s = −0.7, ∆ek = 1.2209.

In short, we can suggest from the above results that ∂LPMf (rf ,m) /∂s < 0 for m =1, 2, 3, ∂LPMf (rf , 1) /∂ek < 0 and ∂LPMf (rf ,m) /∂ek > 0 for m = 2, 3. These resultsare also supported by studying the behaviour of θ2,m and θ3,m in (21) from many simulatedparameters of µ and σ. The simulation results confirm the previous conclusions.16 Thus,it is held that θ2,m < 0, θ3,1 < 0, θ3,2 > 0 and θ3,3 > 0.

5.3 Iso-curves for performance measures

We obtain the points (ek, s) that provide the same value for the selected Kappa measuregiven fixed levels of τ , µ and σ. To shorten, let Ψ denote the vector (µ, σ, τ) and let Ψ0 bea fixed value for Ψ. Thus, the iso-curve associated for any Kappa measure, or iso-Kappa,corresponds to the following set of points Π defined as

Π(m,Ψ0) =

(ek, s) ∈ Γ : Kf (τ0,m) =

µ0 − τ0

m

√LPMn(τ0,m) + s√

3!θ2,m + ek√

4!θ3,m

, (24)

where Kf (τ0,m) denotes a fixed value for the Kappa ratio given by equations (3) and(16). These spaces are easily obtained according to the following Corollary.

Corollary 6 The iso-Kappa (24) implies a linear relation between s and ek. Thus, s =am + ϕm ek such that the slope ϕm is defined in (23) and

am =

√6

θ2,m[ξ0,m − LPMn(τ0,m)] , ξ0,m =

[µ0 − τ0

Kf (τ0,m)

]m, (25)

with LPMn(τ0,m) given in (17).

16We obtain pairs (µi, σi), i ≤ 10, 000 where each parameter is obtained randomly (uniform distribution)such that µi ∈ [0.49%, 0.96%] and σi ∈ [0.96%, 2.61%]. The minimum and maximum values for µ and σare selected from Table 3 in Appendix A. We also choose different values for τ . More details are availableupon request. Finally, see footnote 18.

13

Proof. It is straightforward.

The iso-Kappa in (25) will be labeled as ’iso-Omega-Sharpe’, ’iso-Sortino’ and ’iso-Kappa 3’ respectively for m = 1, 2, 3. Since ∂ξ0,m/∂Kf (τ0,m) < 0 for µ0 > τ0, then∂am/∂Kf (τ0,m) > 0 iff θ2,m < 0. Let Ψ0 = (0.86%, 2.61%, 0.39%) be the parameter setused to obtain Table 1, then the slopes ϕm for the different iso-Kappas (see the values ofϕm in Subsection 5.2) verify that ϕ1 < 0, ϕ2 > 0, ϕ3 > 0. So, an increase in ek leadsto a decrease (increase) in s when moving along the iso-Omega-Sharpe (iso-Sortino oriso-Kappa 3) curve.

By setting s = −0.7 and taking higher levels of ek, we can see in Table 1 thatKf (0.39%, 1) increases (ξ0,1 decreases) but Kf (0.39%,m) decreases (ξ0,m increases) form = 2, 3. This means that the iso-Omega-Sharpe curves with negative slopes move inparalell to the right with higher levels of Kf (0.39%, 1) since a1 in (25) increases becauseθ2,1 < 0. Nevertheless, both the iso-Sortino and iso-Kappa3 curves with positive slopesmove in paralell to the right with lower levels of Kf (0.39%,m) since a1 decreases becauseθ2,m < 0.17

Note that, on the one hand the iso-Kappas from Corollary 6 can be very restrictivesince we are fixing both the mean and volatility parameters for the portfolio returns, buton the other hand we get linear equations which can contribute to a better understandingthe behaviour of the iso-Kappas.18

Finally, outside the Kappa measures, if we get the iso-curves under the FT measures(except for q = m in (19) such as, for instance, the Omega measure for q = m = 1), theydo not hold a linear relation between s and ek. Nevertheless, we could obtain a linearapproximation. For instance, consider the ’iso-Upside potential ratio’ corresponding tothe following set of points Υ defined as:

Υ(Ψ0) =

(ek, s) ∈ Γ : FTRf (τ0, 1, 2) =

µ0 − τ0 + LPMn (τ0, 1) +s√3!θ2,1 +

ek√4!θ3,1

√LPMn (τ0, 2) +

s√3!θ2,2 +

ek√4!θ3,2

,

(26)where Ψ0 is defined previously and FTRf (τ0, 1, 2) denotes a fixed value of the performancemeasure in (19) from Corollary 3. Thus, using a multivariate Taylor expansion of first orderaround the point (s, ek) = (0, 0) in the denominator of FTRf (τ0, 1, 2), we can get a linearrelation between s and ek.

6 Simulation analysis

We implement a simulation analysis based on the closed-form expressions for the perfor-mance measures by assuming the GC distribution for the standardised stock returns in (9).

17The values for θ2,m are, respectively, θ2,1 = −7.53×10−4, θ2,2 = −2.18×10−4 and θ2,3 = −1.87×10−5.The curves for the iso-Kappas are not exhibited here for the sake of brevity.

18Suppose that we do not fix µ in Corollary 6. Then, the iso-Kappas depend on µ, s and ek. By usingthe implicit function Theorem, we can obtain the corresponding partial derivatives (evaluated at a certainpoint) to analyze the behavior of µ with respect to s and ek. This extension is not shown here but it isavailable upon request.

14

The advantage of using the GC distribution is that the simulated value of any performancemeasure exhibitd in Section 2 is obtained immediately. That is, once we obtain randomlya parameter vector (µi, σi, si, eki) representing the return distribution in (9) of a hypothet-ical portfolio i, we do not need to simulate paths of a certain length for monthly returnsto get the alternative performance measures since we have their closed-form expressionsin Section 5. We start our analysis by obtaining the Spearman’s rank correlation betweenthe Sharpe ratio and any alternative performance measure for different portfolios. Thehigher this rank correlation, the lower difference in ranking between the Sharpe ratio andthe selected performance measure. We also study how skewness and excess kurtosis canaffect the evaluation of portfolios.

6.1 Simulation of parameters and performance ratios

We generate randomly the parameter vector (µi, σi, si, eki), i = 1, ..., NT .19 The range of

these parameters is obtained from their CML estimations of these parameters for the hedgefunds,20 see Table 3 in Appendix A. Let xmin (xmax) denote the minimum (maximum) valuefor the CML estimation of the parameter x = σ, s, SR. Then, we have σmin = 0.963%,σmax = 2.163%, smin = −0.798, smax = 0.987, SRmin = 1% and SRmax = 22.3%.

We simulate four independent uniform random variables Uj , j = 1, ..., 4 on the interval(0, 1), each with sample size, N , of 10, 000. The realizations of these variables will bedenoted as uji, i = 1, ..., N . We implement the following steps for i :

1. We compute σi = σmin+(σmax − σmin) u1i and SRi = SRmin+(SRmax − SRmin) u2i.Then, the mean is obtained as µi = rf + σiSRi.

2. The skewness is obtained as si = smin + (smax − smin) u3i. The corresponding ex-cess kurtosis is eki = eki,min + (eki,max − eki,min) u4i such that both (eki,min, si) and(eki,max, si) belong to the restricted space Γ, that is, (eki, si) ∈ Γ.21

3. After simulating ϑi = (µi, σi, si, eki), we compute all the PMs described previouslyby plugging ϑi into the corresponding formulas. To set a fair comparison amongthese measures, we select τ = rf . Thus, given the portfolio i defined by ϑi, weget SRi and any other performance measure, denoted as πi, that can be Ki(rf ,m),FTRi (rf , q,m), RV aRα

i (rf ) or RCV aRαi (rf ).

6.2 Rank correlations

We obtain the average of the Spearman’s correlation over a sample size of one-hundredrank correlations between πi and SRi, such that each correlation is obtained throughN vectors (πi, SRi) computed for portfolios characterised by the vector ϑi according toSubsection 6.1. For instance, we obtain the correlation between FTR (rf , q,m) in (19), for(positive) integer values of q,m ≤ 6, and SR. It is shown (not reported here but availableupon request) that the larger q and m, the lower the rank correlation. If q ≥ 2,m ≥ 3,

19NT = N ×T where as we will see later, N denotes the sample size per regression and T is the numberof regressions. We will set N = 10, 000 and T = 100.

20We rule out HF6 since its Sharpe ratio is negative, see Subsubsection 6.3.1.21We use the notation eki,min and eki,max to emphasize that the range for the possible values of ek

depends on si.

15

the correlation never exceeds 25%. Then, these PMs lead to different portfolio rankingsrespecting the Sharpe ratio. Nevertheless, the Omega or Kappa of order one exhibits avery high correlation (97.70%).22 This evidence suggests that there is no ranking differencewith regard to the Sharpe ratio. All the above results are also supported by Eling et al.(2011) who analyze, among others, the FTRs for different values of q andm. The followingSubsection is about a more robust analysis by splitting the total sample N = 10, 000 intwo subsamples depending on the size of SR. It will be shown, for instance, that therecan be even more ranking difference between K (rf , 1) and SR the higher SR.

6.3 Effects of skewness and kurtosis on portfolio evaluation

Here, we will mainly concentrate on studying those PMs with higher rank correlationsaccording to our previous results in Subsection 6.2. The next analysis is rather similar tothe one by Zakamouline (2011).

6.3.1 The models

We consider the following two models:

1. The first model is defined as

πi = απSRβπ

i , απ, βπ > 0, i = 1, · · · , N, (27)

where π is a specific performance measure (and the same PM for all portfolios iin the above equation) such that πi > 0.23 The portfolio i is defined regarding theparameter vector ϑi. As απ and βπ are positive, πi is equivalent to SRi in the sensethat both produce the same ranking. If testing with data the evidence of a highequivalence between πi and SRi, in that case there would be a high goodness of fitthrough the (adjusted) R2 statistic, denoted by R2

π,0, from estimating by ordinaryleast squares (OLS) the equation (27) in logarithmics.

2. The second model is given as

πi = απSRβπ

i exp(βsπsi + βekπ eki + επ,i

), i = 1, ..., N, (28)

where επ,i is the error term according to π and ϑi. Note that βsπ and βekπ are,respectively, the (relative) sensitivity of π to the skewness and excess kurtosis fromthe portfolio return distribution. That is, βsπ = (∂π/∂s) /π and βekπ = (∂π/∂ek) /π.We estimate by OLS the following expression, obtained after taking logarithms in(28), given by

log (πi) = log (απ) + βπ log (SRi) + βsπsi + βekπ eki + επ,i. (29)

Let R2π,1 denote the (adjusted) R2 statistics of (29). If the estimates of βsπ and βekπ

were statistically significant, then both s and ek can affect π and hence, π could

22The same happens to K (rf , 2) and K (rf , 3) with correlations of, respectively, 94.29% and 91.18% butlower than that of K (rf , 1). For the upside potential ratio, FTR (rf , 1, 2), the rank correlation is 63.23%.Finally, for α = 1%, 5%, the rank correlations for RV aR (RCV aR) are, respectively, 79.11% and 74.35%(87.88% and 79.82%)

23We just concentrate on those portfolios having positive PMs as the relevant ones in our study.

16

produce a different ranking among different portfolios than SR. It also means thatR2

π,1 − R2π,0 would become large and the (Spearman’s) rank correlation coefficient

between π and SR, denoted as RS (π, SR), would be small. Otherwise, if the esti-mates of both parameters were not significant, then R2

π,0 and R2π,1 would be rather

the same value and the above rank correlation would be larger.

6.3.2 Estimation results

Table 2 provides the OLS estimates of βπ, βsπ and βekπ , the rank correlation RS (π, SR)

and the statistics R2π,0 and R2

π,1 obtained from a total of nine regressions from (29). Za-

kamouline (2011) shows24 that a larger SR implies a lower RS (π, SR). A possible rea-son might be the larger the Sharpe ratio the larger the adjustment for non-normality ofthe portfolio return distribution by the selected PM. So, our simulation analysis aimsto test this behaviour by using two non-overlapping ranges for SR for each PM. Specif-ically, we take SRmin and SRmax from Subsection 6.1 and then, consider the intervalsJ1 ≡ [1%, 16.14%] and J2 ≡ (16.14%, 22.3%].25 We split each sample size N in two parts,N1 and N2, and run two regressions with πi as dependent variable. In the first regression,the independent variables are the vectors (1, SRi, si, eki) such that both si and eki comefrom ϑi and SRi = (µi − rf ) /σi ∈ J1 where µi and σi also belong to ϑi. The secondregression is based on the remaining N2 points such that each SRi ∈ J2. The last columnof this Table displays the Chow test (and its p-value).That is, testing the null hypothesisof no structural break (one regression) against the alternative one of structural break (tworegressions). In short, this experiment is repeated 100 times and Table 2 exhibits themean values of the above parameter estimates, rank correlations, etc. The main resultsfrom this Table are as follows:

• R2 statistics, rank correlation and Chow test

In most cases, it is better running two regressions than one regression since the p-valuesfor the Chow test are zero for all PMs except RV aR and RCV aR with α = 1%. In otherwords, the larger SR the larger the sensitivity of any π to the higher moments. For anyπ, the rank correlation RS (π, SR) becomes lower under J2. R

2π,1 − R2

π,0 becomes prettyhigher under J2. In short, these results and (most of) the next ones agree with those ofZakamouline (2011).

• Behaviour of OLS beta estimates

- For all performance measures, the OLS estimates for the three betas, β, arestatistically significant at the 1% level and so, both skewness and excess kurtosisplay significant roles in these measures.

- For any π, βπ is always positive and becomes larger for the regression underJ2, although there is no much difference between both regressions. So, we canconclude that a higher SR leads to a higher π.

24She implements a simulation analysis by assuming the Normal-Inverse-Gaussian for the return distri-bution.

25The number 16.14% is the mean of the Sharpe ratios from nine hedge funds (except HF6).

17

- βsπ is larger in the regression under J2 and always positive suggesting that allmeasures appreciate positive skewness.

- βekπ is negative in most cases except the Omega-Sharpe ratio and RV aR withα = 5%. It suggests that most PMs penalize positive excess kurtosis.

-∣∣∣βsπ∣∣∣ is much greater than

∣∣∣βekπ∣∣∣ in most PMs except the Omega-Sharpe ratio.

[INSERT TABLE 2 AROUND HERE ]

7 Efficient frontiers under LPMs

Each portfolio from the efficient frontier (EF) shows either the highest level of expectedreturn at a given level of risk, measured through the LPM or variance, or the lowest riskfor a given expected return.26 The portfolios include n securities where the marginaldistribution for the standardised stock returns in (9) follow the GC distribution in (10).This Section is structured as follows. First, we present the optimization problem underthis new framework. For simplicity, we assume a multivariate Gaussian copula to capturethe dependence among the stock returns. Second, we compare the portfolios in the EFunder the LPMs with those from the traditional mean-variance (MV) approach. Thiscomparison is interesting since it allows to understand which sort of portfolios, in termsof skewness and excess kurtosis, are selected in the EF frontier according to the LPM riskmeasure in (16).

7.1 Optimization method

The optimization program, OPm, is as follows:

minwLPMp (τ,m) = w′ · CLPM · w, (30)

subject to

w′ · µ = µp, (31)

w′ · l = 1, (32)

w ≥ 0, (33)

where LPMp (τ,m) is the LPM measure of a return portfolio p, w = (w1, . . . , wn)′ and

µ = (µ1, · · · , µn)′ denote, respectively, the weight and expected stock return vectors, andl is the n × 1 vector of ones. Note that we only allow for long positions in the stocks.Finally, CLPM ≡ CLPM (τ,m) in (30) is the co-lower partial moment matrix of ordern × n. To simplify the computations, in a similar way to Nawrocki (1991) and Huang etal. (2001), we assume symmetry in CLPM.27 The elements of the CLPM are defined as

CLPMjj = LPMj (τ,m)2/m , j = 1, · · · , n (34)

CLPMjk = ρjkLPMj (τ,m)1/m LPMk (τ,m)1/m , j 6= k (35)

26We do not obtain EFs under the UPM-LPM framework. This issue is left for future research. SeeCumova and Nawrocki (2014) for more details.

27The asymmetric CLPM matrix is implemented, among others, in Cumova and Nawrocki (2011).

18

where LPMj (τ,m) denotes the LPM in (16) for the stock return j and ρjk represents thecorrelation coefficient between the stock returns j and k. Since CLPM (τ,m) is symmetric,OPm turns to be a quadratic (convex) optimization problem such that each local minimumis a global one. Finally, the traditional variance-covariance Markowitz algorithm, OP0, isthe same as the above OPm but now the objective function (30) is the variance of theportfolio stock return, σ2p. That is, σ

2p = w′M2w where M2 is the covariance matrix.

7.2 MV versus MLPM comparison

Consider again as true parameters the CML estimations of (µ, σ, s, ek) for each hedgefund (see Table 3 in Appendix A). We will implement both OPm and OP0 algorithms tocompare the mean-LPM (MLPM) and the traditional Markowitz efficient frontiers. LetwMV and wMLPM denote the weigth vector of the portfolio included, respectively, in amean-variance (MV) and a MLPM efficient portfolio. As Grootveld and Hallerbach (1999),we measure the difference in the portfolio compositions in both EFs by using the root meansquared dispersion index:

RMSDI (τ,m) =

√√√√ 1

Np × n

Np∑

i=1

n∑

j=1

(wMLPMij − wMV

ij

)2, (36)

where Np is the total number of selected efficient portfolios, n is the number of assetsincluded in a portfolio, and wMV

ij and wMLPMij denote the investment fractions of, respec-

tively, the MV and MLPM efficient portfolio i for the security j. We select the portfolioi such that the expected portfolio returns under the two different risk measures coincide,that is,

∑nj=1w

MVij µj =

∑nj=1w

MLPMij µj .

[ INSERT FIGURE 4 AROUND HERE ]

Figure 4 includes the dispersion index in (36) for the EF portfolios by solving the OPm

algorithm under different values of m and τ , and comparing each with the EF portfoliosprovided by the OP0 algorithm.28 The right-hand graph exhibits, for one hundred EFportfolios (Np = 100) obtained by using the OPm algorithm,29 the values of RMSDI (τ,m)as a function of τ ∈ [0%, 0.39%]. We see large differences between the portfolio compositionunder the selected LPM (τ,m) and MV across the different values of τ . Note that thehigher τ , the lower RMSDI (τ,m) verifying that MLPM portfolios under m = 3 tend toapproach more to the EF portfolio composition under the MV framework. For example,for τ = 0.39%, we have RMSDI (0.39%, 1) = 3.66%, RMSDI (0.39%, 2) = 3.18%, andRMSDI (0.39%, 3) = 2.65%. Our results agree, for example, with those from Marmerand Ng (1993) and Jarrow and Ng (2006) who showed that, for non-Gaussian returndistributions, the mean-semivariance optimal portfolios are significantly different frommean-variance optimal portfolios.

The left-hand graph shows for each m the values of RMSDI (τ,m) as a function ofthe mean of the monthly portfolio return such that the selected return threshold is the

28The correlation matrix is the sample one obtained from the ten HF monthly return series (n = 10)introduced in Appendix A.

29The expected portfolio returns range from µ1 = 0.5182% to µ100 = 0.9553% since Np = 100. Thisinterval is obtained from the intersection of the different ranges for the mean of the portfolio returns fromthe four EFs (m = 1, 2, 3 and MV).

19

monthly risk-free rate, τ = 0.39%. For instance, each point in the Omega-Sharpe line(m = 1) is the root of the mean of the squared errors (wMLPM

ij − wMVij )2, j ≤ 10 of the

portfolio i (where i ≤ 100) in the EF with mean µi (see footnote 7.2). In short, we getthe RMSDI (0.39%, 1) for each portfolio i with mean µi by using (36) with Np = 1. Forany m, we can see significant differences between the portfolio composition under bothLPM (0.39%,m) and MV for different values of µi.

7.3 Skewness and kurtosis of portfolio stock returns

According to the previous assumptions, we will compute the theoretical skewness andexcess kurtosis, denoted as sp and ekp, for the portfolio returns in the different MLPMefficient frontiers. These portfolios include n stocks with GC as the marginal distributionfor each standardised stock return and a multivariate Gaussian copula for the dependenceamong the stock returns.30 Note that we first have to obtain the co-skewness and co-kurtosis matrices, denoted as M3 and M4 respectively.

7.3.1 Co-skewness and co-kurtosis matrices

The co-skewness and co-kurtosis among asset returns are defined31 as

sijk = E [(ri − µi) (rj − µj) (rk − µk)] , i, j, k = 1, . . . , n (37)

andkijkl = E [(ri − µi) (rj − µj) (rk − µk) (rl − µl)] , i, j, k, l = 1, . . . , n. (38)

Hence, the co-skewness matrix of order(n, n2

)is

M3 = [S1jk · · ·Snjk] ,

where Sijk, i = 1, · · · , n denotes the short notation for the (n, n) submatrix (sijk)j,k=1,...,n

with elements in (37).

The co-kurtosis matrix of order(n, n3

)is

M4 = [K11klK12kl . . . K1nkl · · · Kn1klKn2kl . . . Knnkl] ,

where Kijkl, i, j = 1, · · · , n is the short notation for the (n, n) submatrix (kijkl)k,l=1,··· ,nwith elements in (38). Because of certain symmetries, not all the elements of these ma-trices need to be computed. Thus, M3 and M4 involve only

(n+23

)and

(n+34

)elements,

respectively. Since we assume a Gaussian copula, these matrices are easily computed asshown in the next Lemma.

Lemma 1 Let (r1, . . . , rn) denote the random vector of n stock returns such that ri =µi + σizi where zi ∼ GC (0, 1, si, eki) with pdf in (10). The dependence among thesereturns is captured by a Gaussian copula with elements ρij . Then, the elements of M3 in(37) are given by

sijk =

{σ3i si, i = j = k,

0, otherwise,(39)

30See Cherubini et al. (2004) for more details.31We adopt the notation by Jondeau and Rockinger (2006).

20

where σi and si are, respectively, the standard deviation and skewness of the return ri.The elements of M4 in (38) are given as

kijkl =

{σ4i (3 + eki) , i = j = k = l,

σijσkl + σikσjl + σilσjk, otherwise,(40)

where eki is the excess kurtosis of the return ri and σij = σiσjρij denotes the covariancebetween ri and rj.

Proof. See, among others, Isserlis (1918) or Holmsquit (1988) for the higher-order mo-ments of the multivariate normal distribution.

Note that the GC marginal distribution is easily imposed in the above equations (39)and (40) verifying that siii = σ3i s

3i and kiiii = σ4i (3 + eki). Finally, the skewness and

kurtosis of the portfolio stock return are obtained as

sp =w′M3 (w ⊗ w)

σ3p, (41)

and

kp =w′M4 (w ⊗ w ⊗ w)

σ4p, (42)

where ⊗ denotes the “Kronecker product”. Finally, ekp = kp − 3 is the portfolio excesskurtosis.

7.3.2 Getting sp and ekp from MLPM efficient frontiers

Figure 5 exhibits sp and ekp for the one hundred EF portfolios and return thresholdsτ = 0%, 0.39% as a function of the monthly expected return µj ∈ [0.5182%, 0.9553%].32

The left-hand graphics include the values of sp under τ = 0% (upper graph) and τ = 0.39%(lower graph) while the values of ekp are shown on the right side.33 Several results arise.First, for any τ and m, there are differences beween MV and MLPM. These differencesbecome more important for m = 1. Second, there are no significant differences betweenLPM (τ, 2) and LPM (τ, 3). Third, there are significant differences between LPM (0%, 1)and LPM (0%,m) where m = 2, 3 for lower expected portfolio returns.

[ INSERT FIGURE 5 AROUND HERE ]

8 The SNP density and LPMs

We show here the SNP density already introduced in Subsection 3.1 to model stock returns.The main advantages of this density are the following. First, it nests the GC distributionin (10), and so, it allows a higher flexibility in terms of skewness and excess kurtosis.Second, it is always positive and no restrictions on its parameters are needed to guaranteethe positivity of the pdf. Third, it provides easily closed-form expressions for the LPMs

32This interval is the same as that used to obtain the dispersion index in Figure 4.33Note that, in the MV setup, both sp and ekp in the upper graphs are the same as in the lower ones

since OP0 does not depend on τ .

21

since we can apply some properties already obtained under the GC density. A possibledrawback respecting the GC density is that both skewness and excess kurtosis are notdirectly the SNP parameters but a function of them. In short, the SNP density of arandom variable x is defined as

h (x) =φ (x)

v′v

(p∑

i=0

viHi (x)

)2

, (43)

where v = (v0, v1, ..., vp)′ ∈ R

p+1, φ (·) denotes the pdf of a standard normal variable andHi (x) are the normalized Hermite polynomials in (11). Since h (·) is homogeneous ofdegree zero in v, we can either impose v′v = 1 or v0 = 1 to solve the scale indeterminacy.By expanding the squared expression in (43), we arrive at Proposition 1 in Leon et al.(2009). For p = 2, we get

h (x) = φ (x)

4∑

k=0

γk (v)Hk (x) , (44)

where

γ0 (v) = 1, γ1 (v) =2v1(v0+

√2v2)

v′v , γ2 (v) =√2(v21+2v2

2+√2v0v2)

v′v ,

γ3 (v) =2√3v1v2v′v , γ4 (v) =

√6v2

2

v′v .(45)

We are interested in an affine transformation z∗ = a (v) + b (v) x with g (·) as thedensity of z∗ verifying that Eg [z

∗] = 0, Eg

[z∗2]= 1. Hence, the location parameter a (v)

and the scale one b (v) are obtained as

a (v) = − Eh [x]√Vh [x]

, b (v) =1√Vh [x]

, (46)

where Eh [x] and Vh [x] denote, respectively, the mean and variance of x with h (·) in (44)as pdf. Finally, we can express the stock return r as

r = µ+ σz∗ = µ+ aσ + bσx, (47)

such that f (r) = h (x) / (bσ) is the pdf of r. The mean and variance of r are, respectively,Ef [r] = µ and Vf [r] = σ2. The next Proposition shows the general LPM expression of rwith the SNP density.

Proposition 4 Let r be the stock return in (47) with pdf defined as f (r) = h (x) / (bσ)such that h (x) is the SNP density in (44). The lower partial moment LPMf (τ,m) of rcan be expressed as

LPMf (τ,m) =m∑

j=0

(−1)j(m

j

)κm−j0 κj1Cj, (48)

where κ0 = τ − µ− aσ, κ1 = bσ, the parameters a and b are defined in (46) and

Cj =4∑

i=0

ξi (v)Bj+i, (49)

22

such that Bl =∫ τ+−∞ xlφ(x)dx, with general solution in (64) from Appendix B, where τ+ =

κ0/κ1 and ξi (v) is given by the following expressions:

ξ0 (v) = 1− γ2 (v) /√2 + 3γ4 (v) /

√4!,

ξ1 (v) = γ1 (v)− 3γ3 (v) /√3!,

ξ2 (v) = γ2 (v) /√2− 6γ4 (v) /

√4!, (50)

ξ3 (v) = γ3 (v) /√3!,

ξ4 (v) = γ4 (v) /√4!,

where the coefficients γk (v) are defined in (45).

Proof. See Appendix B.

Finally, we could also obtain the efficient frontiers by assuming now a different SNPpdf as marginal distribution for each stock return. Thus, a different parameter vector vin (44) for each stock return.

9 Conclusions

We have obtained the closed-form formulas for the partial moments, that is, both upper(UPMs) and lower partial moments (LPMs) when the implied distribution for the stockreturns is driven by the Gram-Charlier (GC) density with restrictions on the skewness,s, and excess of kurtosis, ek, parameters to guarantee the probability density function(pdf) is well-defined, see Jondeau and Rockinger (2001). We can express the UPMs asfunctions of the LPMs. It is verified that the LPMs under GC become linear functionson both s and ek. Because of this, we can easily understand the behaviour of this kindof downside-risk measures with respect to changes in s and ek and hence, the behaviourof the related performance measures (PMs). These PMs are mainly the Farinelli-Tibilettiratios (FTRs) and the Kappa measures. We also study the PMs based on the Value atRisk and Expected Shortfall (or CVaR) under the GC density.

Our simulation study concludes that the choice of the performance measure (PM)can affect the evaluation of portfolios differently to the Sharpe ratio (SR) because of thesensitivity of the selected PM to the levels of s and ek implied in the portfolio returns. Ourresults agree with those from Zakamouline (2011). Thus, the selection of PMs becomesrelevant in ranking portfolios.

We have also obtained the efficient frontiers (EFs) based on the LPMs as an alternativerisk measure taking the standard deviation as benchmark. We have compared the portfoliocomposition from the EFs under both the mean-LPM and mean-variance settings. We haveshown large differences in both compositions through a sensitivity analysis by changingthe return threshold. As a result, we can conclude that the choice of the risk measureaffects the portfolio composition from the EF. Finally, we have also found some differencesamong the skewness and excess kurtosis levels implied in the portfolios from these EFs.

Several issues are left for further research. First, we can study the portfolio optimiza-tion but using LPMs under alternative dependence structure between financial returns

23

using alternative copula functions with marginal stock return distributions capturing theautocorrelation and GARCH structure. Boubaker and Sghaier (2013) show this kindof analysis but restricted to the mean-variance EFs setup under the presence of long-memory for daily hedge fund returns. Harris and Mazibas (2013) consider alternativeoptimization frameworks such as the Mean-CVaR, Omega optimization model under bothAR(1)-GARCH family and copula modeling for monthly hedge fund returns. Finally, itwould be interesting to study the optimal combination of performance measures, see Billioet al. (2012).

24

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28

Appendix A. Data and ML estimation

The ten hedge fund (HF) indices used here are available from www.hedgefundresearch.com.The same dataset has been used in Mausser et al. (2006) to optimise a portfolio’s omegaratio based on a linear programming method. We collect monthly index HF returns forthe period from November 1999 to February 2008 with a total of 100 observations. Table3 reports the constrained maximum likelihood (CML) parameter estimates and the MLstandard errors for these monthly return series by assuming that the data come from (9).Thus, rj,t = µj + σjzt where zt ∼ iid GC(0, 1, sj , ekj), j = 1, . . . , 10 represents the hedgefund HFj and t = 1, ..., 100. The names implied by the index symbols (starting with”HFRX”) in the first row of the Table are the following: HFRXGL (Global Hedge Fund),HFRXEW (Equal Weighted Strategies), HFRXCA (Convertible Arbitrage), HFRXDS(Distressed Securities), HFRXEH (Equity Hedge), HFRXEMN (Equity Market Neutral),HFRXED (Event-Driven), HFRXM (Macro Index), HFRXMA (Merger Arbitrage) andHFRXRVA (Relative Value Arbitrage). The sample correlation matrix for the ten monthlyreturn series of HFs for the above period is not exhibitd for the sake of brevity. It is shownthat all sample correlations are positive. The maximum correlation is between HF1 andHF5 (0.887) and the minimum one is between HF6 and HF8 (0.019).

[INSERT TABLE 3 AROUND HERE ]

29

Appendix B. Proofs

Proof of Proposition 1

Kendall and Stuart (1977) shows that

Hk (z) =

[k/2]∑

n=0

ak,n zk−2n, (51)

where [·] rounds its argument to the nearest (smaller) integer and

ak,n =

(−1

2

)n √k!

(k − 2n)!n!.

Taking expectations in (51) with pdf g (·), see equation (10), we obtain

Eg [Hk (z)] =

[k/2]∑

n=0

ak,nEg

[zk−2n

]. (52)

Given (10) and (12), we get Eg [Hk (z)] = 0 for k ≥ 5 and so, we obtain recursivelythe expression for Eg

[zk]in (13). For instance, setting k = 5 and plugging Eg [z] = 0

andEg

[z3]= s into (52), we get Eg

[z5]= 10s. For k = 6, we have Eg

[z6]= 15 + 15ek,

etc. Finally, regarding the moments of r in (9), with pdf f (r) = g (z) /σ, we obtain (14)by using the binomial expansion.

Proof of Proposition 2

Let z be the standardised return of r in (9) with pdf g (z) in (10). Its distributionfunction is

FGC (z; s, ek) =

∫ z

−∞g (x) dx =

∫ z

−∞p(x)φ (x) dx

= Φ(z) +s√3!

∫ z

−∞H3 (x)φ (x) dx+

ek√4!

∫ z

−∞H4 (x)φ (x) dx

= Φ(τ∗)− s

3√2!H2 (τ

∗)φ (τ∗)− ek

4√3!H3 (τ

∗)φ (τ∗) . (53)

where the last equality arises from the following relationship (see Leon et al. (2009) fordetails): ∫ a

−∞Hk(x)φ(x)dx = − 1√

kHk−1(a)φ(a), k ≥ 1. (54)

Let f (r) = g (z) /σ be the pdf of r. Noting that LPMf (τ, 0) = FGC (τ∗; s, ek) whereτ∗ = (τ − µ) /σ completes the proof.

Proof of Corollary 2

We can rewrite (15) as

LPMf (τ, 0) = B0 +s√3!A02 +

ek√4!A03 (55)

30

where

B0 = Φ(τ∗), A02 = − 1√3H2(τ

∗)φ(τ∗), A03 = − 1√4H3(τ

∗)φ(τ∗), (56)

It holds that B0 > 0 ∀τ∗, A02 > 0 iff |τ∗| < 1 and A03 > 0 iff τ∗ ∈(−∞,−

√3)∪(

0,√3). Moreover, using the relations (a) dφ (x) /dx = −xφ (x) and (b) dHk (x) /dx =√

kHk−1 (x), we get

∂A02

∂τ∗> 0 ⇔ τ∗ ∈

(−√3, 0)∪(√

3,+∞),

∂A03

∂τ∗> 0 ⇔ τ∗ ∈ (−∞,−τ∗1 ) ∪ (−τ∗2 , τ∗2 ) ∪ (τ∗1 ,+∞) ,

where τ∗1 =√

3 +√6 and τ∗2 =

√3−

√6. Hence,

∂LPMf (τ, 0)

∂s> 0 ⇔ |τ∗| < 1,

∂LPMf (τ, 0)

∂ek> 0 ⇔ τ∗ ∈

(−∞,−

√3)∪(0,√3),

and ∂LPMf (τ, 0)/∂τ∗ > 0 if τ∗ ∈ (−τ∗2 , 0) ∪ (τ∗1 ,+∞). Finally, the signs of the partial

derivatives of LPMf (τ, 0) with respect to τ , µ and σ are obtained by applying the chainrule and using that ∂τ∗/∂τ > 0, ∂τ∗/∂µ < 0 and ∂τ∗/∂σ > 0 iff µ > τ .

Proof of Proposition 3

Let f and g denote, respectively, the pdfs for r and z in (9). So, it is verified thatf (r) = g (z) /σ where g (·) denotes the GC pdf and z = (r − µ) /σ. Then, we can rewrite(1) as

LPMf (τ,m) =

∫ τ

−∞(τ − r)mf(r) dr =

∫ τ∗

−∞(τ − µ− σz)mg(z) dz (57)

where τ∗ = (τ − µ) /σ. If we apply the binomial expansion on (τ −µ− σz)m in (57), then

LPMf (τ,m) =∑m

k=0(−1)k

(m

k

)(τ − µ)m−k σkIk, (58)

where Ik=Eg

[zk |z < τ∗

]denotes the conditional expected value. Thus, for k ≥ 1 we

obtain

Ik =

∫ τ∗

−∞zkg(z) dz = Bk +

s√3!Ak2 +

ek√4!Ak3, (59)

where

Bk =

∫ τ∗

−∞zkφ(z)dz, Ak2 =

∫ τ∗

−∞zkH3(z)φ(z)dz, Ak3 =

∫ τ∗

−∞zkH4(z)φ(z)dz. (60)

Both Ak2 and Ak3 can be expressed, by using (54), as

Ak2 =1√3!(Bk+3 − 3Bk+1), Ak3 =

1√4!(Bk+4 − 6Bk+2 + 3Bk). (61)

31

Note that I0 is just the equation of LPMf (τ, 0) in (55) with B0, A02, and A03 given in(56). By plugging (59) into (58), we get

LPMf (τ,m) = LPMn(τ,m) +s√3!θ2,m +

ek√4!θ3,m,

where

θj,m =∑m

k=0(−1)k

(m

k

)(τ − µ)m−k σkAkj ,

where LPMn(τ,m)=En[(τ −r)m+ ] such that n (·) is the pdf of the normal distribution withµ and σ, respectively, as the mean and standard deviation. Thus,

LPMn(τ,m) =

∫ τ∗

−∞(τ − µ− σz)m φ (z) dz =

∑m

k=0(−1)k

(m

k

)(τ − µ)m−k σkBk. (62)

Finally, to get the expression for Bk in (60), we need previously the following result:

zk =

[k/2]∑

n=0

ck,nHk−2n (z) , (63)

where ck,n ∈ R and Hi (·) is a Hermite polynomial. Note that (63), that is available uponrequest, is just the inversion formula of (51). If we take (54) and (63), then

Bk =

∫ τ∗

−∞zkφ(z)dz =

[k/2]∑

n=0

ck,n

∫ τ∗

−∞Hk−2n (z)φ(z)dz

=

[k/2]∑

n=0

ck,n

{Φ(τ∗)1(k−2n=0) +

1√k − 2n

Hk−2n−1 (τ∗)φ (τ∗) 1(k−2n≥1)

}, (64)

where 1(·) is the usual indicator function.

Proof of Corollary 4

The expressions of θj,m in (21) and LPMn(τ,m) in (22) are easily obtained for m ≤ 3by using, respectively, the equations (18) and (17) from Proposition 3. Since we need toobtain Ak2 and Ak3 in (61) for k = 0, 1, 2, 3, 4, then we must previously get Bj in (60) forj = 1, ..., 7. By applying the condition (54),

B1 = −φ (τ∗) ,B2 = −H1 (τ

∗)φ (τ∗) + Φ (τ∗) ,

B3 = −√2H2 (τ

∗)φ (τ∗)− 3φ (τ∗) ,

B4 = −√3!H3 (τ

∗)φ (τ∗)− 6H1 (τ∗)φ (τ∗) + 3Φ (τ∗) ,

B5 = −√4!H4 (τ

∗)φ (τ∗)− 10√2H2 (τ

∗)φ (τ∗)− 15φ (τ∗) ,

B6 = −√5!H5 (τ

∗)φ (τ∗)− 15√3!H3 (τ

∗)φ (τ∗)− 45H1 (τ∗)φ (τ∗) + 15Φ (τ∗) ,

B7 = −√6!H6 (τ

∗)φ (τ∗)− 21√4!H4 (τ

∗)φ (τ∗)− 105√2H2 (τ

∗)φ (τ∗)− 105φ (τ∗) .

Note that the above expressions agree with the general formula for Bj in (64).

Proof of Proposition 4

32

Let h and f denote, respectively, the pdfs for x and r in (44) and (47). So, it is verifiedthat f (r) = h (x) /bσ. Then, we can rewrite (1) as

LPMf (τ,m) =

∫ τ

−∞(τ − r)mf(r) dr =

∫ τ+

−∞(κ0 − κ1x)

mh(x) dx, (65)

where κ0 = τ−µ−aσ, κ1 = bσ > 0 and τ+ = κ0/κ1. The expressions of a and b in (46) canbe obtained from the first two unconditional moments of x, denoted as µ′x (1) and µ

′x (2),

from Lemma 1 in Leon et al. (2009). Thus, Eh [x] = µ′x (1) is just γ1 (v) in equation (45)and

µ′x (2) =2(v21 + 2v22 +

√2v2v0

)

v′v+ 1,

then Vh [x] = µ′x (2)− µ′x (1)2. If we apply the binomial expansion on (κ0 − κ1x)

m in (65)and consider (44), we have

LPMf (τ,m) =

∫ τ+

−∞

[∑m

j=0(−1)j

(m

j

)κm−j0 κj1x

j

]h(x) dx

=∑m

j=0(−1)j

(m

j

)κm−j0 κj1

[Bj +

∑4

k=1γk (v)Gjk

](66)

where Bj =∫ τ+−∞ xjφ(x)dx, with general solution in (64), and Gjk =

∫ τ+−∞ xjHk(x)φ (x) dx.

If we consider (51), then the expressions of Gjk, for 0 ≤ j ≤ m and 1 ≤ k ≤ 4, are:

Gj1 = Bj+1, Gj2 =1√2(Bj+2 −Bj) ,

Gj3 =1√3!(Bj+3 − 3Bj+1), Gj4 =

1√4!(Bj+4 − 6Bj+2 + 3Bj).

(67)

Note that Bl in the above expressions are evaluated at point τ+. By plugging (67) into(66), we finally arrive at the expressions (48)− (50) in Proposition 4.

33

Appendix of Tables

Table 1: Sensitivity analysis for the Kappa measures under GC distribution. The effectsof skewness (s) and excess kurtosis (ek)

Panel A: Omega Panel B: Sortino Panel C: Kappa 3

ek s = −0.7 s = 0 s = 0.4 s = −0.7 s = 0 s = 0.4 s = −0.7 s = 0 s = 0.4

0.8996 0.5801 0.5959 0.6054 0.2622 0.2923 0.3150 0.1825 0.2070 0.2292

1.2048 0.5893 0.6057 0.6154 0.2617 0.2916 0.3141 0.1806 0.2040 0.2246

1.5100 0.5988 0.6157 0.6258 0.2612 0.2909 0.3133 0.1788 0.2011 0.2204

1.8153 0.6086 0.6261 0.6365 0.2607 0.2903 0.3124 0.1771 0.1983 0.2165

2.1205 0.6187 0.6368 0.6476 0.2602 0.2896 0.3116 0.1754 0.1957 0.2128

2.4257 0.6292 0.6479 0.6591 0.2598 0.2889 0.3107 0.1738 0.1932 0.2094

2.7309 0.6400 0.6594 0.6710 0.2593 0.2882 0.3099 0.1723 0.1909 0.2062

3.0361 0.6513 0.6713 0.6833 0.2588 0.2876 0.3091 0.1708 0.1887 0.2031

3.3414 0.6629 0.6836 0.6961 0.2583 0.2869 0.3083 0.1693 0.1865 0.2003

3.6466 0.6749 0.6964 0.7094 0.2578 0.2863 0.3074 0.1680 0.1845 0.1976

This table exhibits the values of the closed-form formulas for the Kappa measures (Omega-Sharpe, Sortino and Kappa3) by using the LPM expressions from Corollary 4 for monthly returns. All portfolios in the table have µ = 0.86% andσ = 2.61% but different values for s and ek such that (ek, s) ∈ Γ. The return threshold is rf = 0.39%. The Sharperatio is constant over this table, SR=0.1796.

34

Table 2: Some results from regression analyses by using simulated monthly returns from GC distribution

log (Perf. Measure) Sharpe ratio interval beta log(Sharpe) beta Sk beta Ek Rank corre R2 R20 Chow Test (p-value)

Omega-Sharpe [1%, 16.14%] 1.1888 0.0286 0.0536 0.9509 0.9996 0.9000 10484.4060

(16.14%, 22.3%] 1.2855 0.0458 0.0570 0.8965 0.9993 0.7941 (0.0000)

Sortino [1%, 16.14%] 1.0993 0.1672 -0.0046 0.8728 0.9984 0.7541 1012.2615

(16.14%, 22.3%] 1.1480 0.1820 -0.0077 0.7489 0.9966 0.5626 (0.0000)

Kappa 3 [1%, 16.14%] 1.0685 0.1974 -0.0417 0.8059 0.9916 0.6439 95.2884

(16.14%, 22.3%] 1.1017 0.2060 -0.0457 0.6583 0.9853 0.4430 (0.0000)

Upside Potential [1%, 16.14%] 0.2665 0.1473 -0.0421 0.3738 0.9938 0.1559 2166.0951

(16.14%, 22.3%] 0.3895 0.1552 -0.0413 0.3588 0.9925 0.1443 (0.0000)

RVaR (1%) [1%, 16.14%] 1.0561 0.2766 -0.1025 0.6272 0.8639 0.3630 4.0309

(16.14%, 22.3%] 1.0844 0.2849 -0.1058 0.5013 0.8291 0.2058 (0.0752)

RVaR (5%) [1%, 16, 14%] 1.1098 0.3826 0.0382 0.5453 0.8648 0.3142 11.1504

(16.14%, 22.3%] 1.1736 0.4028 0.0412 0.4018 0.8373 0.1751 (0.0000)

RCVaR (1%) [1%, 16, 14%] 1.0458 0.1643 -0.0883 0.7544 0.8883 0.5311 3.5877

(16.14%, 22.3%] 1.0692 0.1684 -0.0906 0.6305 0.8418 0.3396 (0.1050)

RCVaR (5%) [1%, 16, 14%] 1.0693 0.3272 -0.0645 0.6125 0.9599 0.3902 16.3899

(16.14%, 22.3%] 1.1062 0.3387 -0.0668 0.4590 0.9488 0.2269 (0.0000)

This table exhibits the results of running by OLS the model given by (29). All regressions contain the same explanatory variables: the constant,the skewness (s), the excess of kurtosis (ek) and the log of the Sharpe ratio (log(SR)). The beta estimates corresponding to s, ek and log(SR) areshown, respectively, in columns 3, 4, and 5. Consider a certain dependent variable (performance measure) in column 1, then the regression is runtwice. The total sample size, N=10,000, is divided in two parts having one regression per subsample. The criterion followed to split N depends onthe size of the Sharpe ratio (SR) of portfolio i ∈ {1, · · · , N}, denoted as SRi. The column 2 is labeled as the SR interval, with elements J1 and J2.If SRi ∈ J1 ≡ [1%, 16.14%], then portfolio i belongs to subsample 1, otherwise it belongs to subsample 2 such that SRi ∈ J2 ≡ (16.14%, 22.3%].This table shows a total of 20 different regressions. The column 6 represents the Spearman’s rank correlation between any performance measurefrom column 1 and the SR for each subsample. Columns 7 and 8, labeled as R2 and R20, correspond, respectively, to the R2 statistic of (29) andthe log of (27). The last column represents both the value and the p-value (in parenthesis) for the Chow test with null hypothesis of no structuralbreak (one regression) against the alternative one of structural break (two regressions). All portfolios, each defined by the vector (µi, σi, si, eki)where i = 1, · · · , N , are simulated according to the procedure in Subsection 6.1. Note that we repeat the above process a total of 100 times, soany value or estimation of the table (from columns 3 to 9) is really the mean over 100 estimates. The beta estimates (columns 3 to 5) are allstatistically significant at the 1% level for all the alternative performance measures.

35

Table 3: CML estimation of monthly return series from HFRX Tradeable Indices

HFRXGL HFRXEW HFRXCA HFRXDS HFRXEH HFRXEMN HFRXED HFRXM HFRXMA HFRXRVA

Mean (%) 0.702 0.605 0.533 0.671 0.955 0.229 0.699 0.859 0.491 0.554

SE 0.162 0.095 0.138 0.149 0.236 0.098 0.145 0.266 0.100 0.098

Std. Dev. (%) 1.656 0.963 1.199 1.445 2.575 0.972 1.518 2.613 1.011 0.998

SE 0.154 0.080 0.123 0.126 0.391 0.082 0.121 0.227 0.085 0.092

Skewness 0.360 -0.275 -0.798 0.408 0.987 -0.280 -0.571 0.401 -0.594 -0.233

SE 0.462 0.278 0.533 0.272 0.416 0.274 0.249 0.316 0.261 0.351

Exc. Kurt. 2.526 0.954 1.121 1.510 3.038 1.219 1.254 1.519 1.508 1.700

SE 0.662 0.689 1.096 0.689 1.320 0.660 0.628 0.599 0.646 0.708

This table shows the constrained maximum likelihood (CML) estimates for the different monthly hedge fund (HF) return series according to (9), thatis, the standardised stock returns are driven by the Gram-Charlier (GC) density. The period goes from November 1999 to February 2008, i.e. a totalsample size of 100 observations per each data series. The CML estimates for the mean and standard deviation parameters are given in percentages.The standard errors (SE) are calculated by using the Quasi-Maximum likelihood (QML) method.36

Appendix of Figures

0 0.5 1 1.5 2 2.5 3 3.5 4−1.5

−1

−0.5

0

0.5

1

1.5Gram−Charlier restricted parameter space for (ek,s)

Excess kurtosis(ek)

Ske

wne

ss(s

)

Figure 1: Space containing for stock returns the points (ek, s) with the “excess kurtosis” level,

ek, in the x-axis and the “skewness” level, s, in the y-axis. This space is limited by a frontier

(envelope) verifying that the Gram-Charlier (GC) density is well defined for the points on and

inside the envelope. Thus, the GC density will be restricted to this space for (ek, s). Note that

ek ∈ [0, 4] while s ∈ [−1.0493, 1.0493]. The range of s depends on the level of ek. See Jondeau and

Rockinger (2001) for more details about how to obtain this frontier.

37

0 0.5 1 1.5 2 2.5 3 3.5 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

s= − 0.7 (τ = −5.49%)

s= 0 (τ = −5.49%)

s= 0.4 (τ = −5.49%)

s= − 0.7 (τ = −2.97%)

s= 0 (τ = −2.97%)

s= 0.4 (τ = −2.97%)

Shortfall probabilities for different (ek,s) and threshold levels (τ = VaR)

ek

shor

tfall

prob

abili

ty

Figure 2: Each line contains the points (ek, LPM(τ, 0)) where ek is the “excess kurtosis” level

of the monthly stock return r and LPM(τ, 0) is the “shortfall probability” in (15). Thus, r is

given in (9) with the standardised stock return, z, distributed under the GC density in (10). Note

that across each line the skewness level, s, is fixed and also the return threshold, τ . Specifically,

τ is just the 5% (1%)-quantile under the GC density. Thus, τ is the VaR and the skewness levels

are s = −0.7, 0, 0.4. Note that the length of each line is different since each point must belong to

the set exhibited in Figure 1. The mean and standard deviation for any monthly stock return are

µ = 0.86% and σ = 2.61%. For the behaviour of these lines, see Corollary 2.

0 0.5 1 1.5 2 2.5 3 3.5 40.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

s= − 0.7 (τ = 0%)

s= 0 (τ = 0%)

s= 0.4 (τ = 0%)

s= − 0.7 (τ = 0.39%)

s= 0 (τ = 0.39%)

s= 0.4 (τ = 0.39%)

Shortfall probabilities for different (ek,s) and nonnegative threshold levels (τ)

ek

shor

tfall

prob

abili

ty

Figure 3: This figure is the same as Figure 2 but here the return thresholds are non-negative.

These values are τ = 0% and τ = 0.39%, where the last one is just the monthly risk-free interest

rate on 10-year US Treasury bonds (i.e., the sample mean from November 1998 to December 2008).

Each line contains the points (ek, LPM(τ, 0)) where ek is the “excess kurtosis” level of the monthly

stock return r, and LPM(τ, 0) is the “shortfall probability” in (15). Thus, r is given in (9) with

the standardised stock return, z, distributed under the GC density in (10). By setting a level for τ ,

we show three lines corresponding to the skewness levels are s = −0.7, 0, 0.4. Note again that the

length of each line is different since each point (ek, s) must belong to the set exhibited in Figure

1. The mean and standard deviation for any monthly stock return are µ = 0.86% and σ = 2.61%.

For the behaviour of these lines, see Corollary 2.

38

0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5RMSDI for each EF portfolio for τ = 0.39%

Monthly portfolio expected return (%)

RM

SD

I (%

)

0 0.1 0.2 0.3 0.4

3

4

5

6

7

8

9

10

11

12

13Total RMSDI where τ goes from 0% to 0.39%

Return threshold, τ (%)

RM

SD

I (%

)

Omega−SharpeSortinoKappa 3

Omega−SharpeSortinoKappa 3

Figure 4: This figure contains two graphs. The left-hand graph shows one hundred efficient frontier

(EF) portfolios by solving the optimization program driven by (30) to (33) for each m = 1, 2, 3

and also, the case under the MV approach with sample covariance matrix obtained from the hedge

fund (HF) data in Appendix A. Other parameters such as (µj , σj , sj , ekj) for each HF return series

(j = 1, · · · , 10) are exhibited in Table 3 in the same Appendix. The return threshold is τ = 0.39%

(monthly risk-free interest rate). The x-axis shows a grid for the mean of monthly portfolio returns

going from 0.5182% to 0.9553%, see Subsection 7.2 for more details. The y-axis displays one

graph for a different m containing the RMSDI, given by (36) with Np = 1, in percentage for each

EF portfolio. The right-hand graph exhibits three lines but now the x-axis shows a grid for the

monthly return thresholds such that τ ∈ [0%, 0.39%]. Meanwhile, the y-axis contains the RMSDI

with Np = 100 in (36). Thus, the case of τ = 0.39% in the left-hand graph corresponds to the last

3 points further on the right whatever the selected line (depending on m).

39

0.5 0.6 0.7 0.8 0.9−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Monthly portfolio expected return (%)

port

folio

ret

urn

skew

ness

Return threshold, τ=0%

0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

Monthly portfolio expected return (%)

port

folio

ret

urn

exce

ss k

urto

sis

Return threshold, τ=0%

0.5 0.6 0.7 0.8 0.9−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Monthly portfolio expected return (%)

port

folio

ret

urn

skew

ness

Return threshold, τ=0.39%

0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

Monthly portfolio expected return (%)

port

folio

ret

urn

exce

ss k

urto

sis

Return threshold, τ=0.39%

MVLPM(0%,1)LPM(0%,2)LPM(0%,3)

MVLPM(0%,1)LPM(0%,2)LPM(0%,3)

MVLPM(0.39%,1)LPM(0.39%,2)LPM(0.39%,3)

MVLPM(0.39%,1)LPM(0.39%,2)LPM(0.39%,3)

Figure 5: This figure provides the skewness (sp) and kurtosis excess (ekp) for the four series of EF

portfolios as a function of the monthly portfolio expected return. The grid is the same as in Figure

4. The values of sp and ekp have been obtained by using, respectively, (41) and (42). Each graph

includes four lines that show, respectively, the results under the MV framework and the LPM

measures LPM(τ,m), m = 1, 2, 3, where τ denotes the monthly threshold return. The upper

and lower graphs include the results for, respectively, τ = 0% and τ = 0.39%. The co-skewness

and co-kurtosis matrices, M3 and M4, are given in closed-form assuming a Gaussian copula with

sample correlation matrix from hedge fund (HF) data in Appendix A. The parameters of the GC

pdf for the marginal distribution of each HF monthly return are given in Table 3 in Appendix A.

40